Timeline for Does quantum cohomology have an $E_\infty$-ring structure?
Current License: CC BY-SA 4.0
13 events
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| Feb 19, 2021 at 21:58 | comment | added | Jonny Evans | @QuantumRing: If I get time, I will, though I don't think my comment answered the question! In the meantime, I can direct you to another answer I wrote to a related question mathoverflow.net/a/352822/10839 | |
| Feb 19, 2021 at 6:09 | comment | added | QuantumRing | @JonnyEvans Would you consider posting an expanded version of your comments as an answer? It looks really interesting but I don't have the right background for the paper you posted. Thanks! | |
| Feb 18, 2021 at 18:02 | comment | added | Bbb | Thanks @JonnyEvans, will try to parse this | |
| Feb 17, 2021 at 23:29 | comment | added | Jonny Evans | @Bbb: I thought about this with YankI Lekili when we were working on this paper: arxiv.org/abs/1507.05842 I don't remember finding anywhere the version with spheres is written, but it would still be pretty Floer theoretic. You might find the above paper a good place to start: there's not much technical Floer theory, just lots of playing with twisted complexes. Section 7.3 has some examples where we show QH is not formal (e.g. for toric Fanos whose QH is not semisimple, or even for P^1 if you work with coefficients in Z/2). | |
| Feb 17, 2021 at 21:11 | comment | added | Bbb | Thanks @JonnyEvans, exactly the type of answer I was looking for - can you point me to a reference for this? (Preferably the quantum cohomology version as I’m an algebraic geometer and not very fluent in Floer theory) | |
| Feb 17, 2021 at 8:18 | comment | added | Jonny Evans | The A_\infty structure comes from thinking of QH as the Lagrangian Floer cohomology of the diagonal. You can phrase it in a more "quantum cohomologyish" language (counting spheres with point constraints which are required to live along an equator in the domain) but this is really the same thing. | |
| Feb 16, 2021 at 19:22 | comment | added | Bbb | Interesting, this makes me wanna ask a simpler question: is there a geometric description of the E_1 structure on quantum cohomology that doesn’t require passing through mirror symmetry? | |
| Feb 16, 2021 at 18:15 | history | edited | QuantumRing | CC BY-SA 4.0 |
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| Feb 15, 2021 at 15:54 | comment | added | Dustin Clausen | It's always bold to claim that two mathematical objects are not related! Anyway, at least Tim's conclusion is correct: according to my understanding, quanutum cohomology should have an $E_2$ structure but not an $E_\infty$ structur in general. This is because quantum cohomology is the Hochschild cohomology of the Fukaya category (derived endomorphisms of the identity functor), and as the Fukaya category only a category not a tensor category this only gets an $E_2$-structure. (Experts please correct me if I'm wrong) | |
| Feb 15, 2021 at 15:40 | comment | added | Sam Hopkins | Quantum cohomology is not really related to quantum groups at all. | |
| Feb 15, 2021 at 14:58 | comment | added | Tim Campion | I don't know anything about quantum cohomology, but if it's anything like quantum groups (about which I also know very little), you might expect to get an $E_2$ structure or something but not an $E_\infty$-structure. | |
| Feb 15, 2021 at 14:07 | review | First posts | |||
| Feb 15, 2021 at 14:25 | |||||
| Feb 15, 2021 at 14:00 | history | asked | QuantumRing | CC BY-SA 4.0 |